I Can you change a planet's orbit by reducing its velocity?

My question is, take Mars, say, and install a huge rocket engine on the ground pointing out the planet (obviously) and in the opposite direction of its velocity, so that if it is turned on, the planet is slowed.
The rocket starts to work and Mars is slowed. If the loss of velocity goes little by little, could we make Mars follow another stable orbit or would it just fall to the Sun?

If you can go more in depth: if the answer to the question asked before is yes, what equation tells me exactly how little the loss of velocity must be in order to keep the planet through another stable orbit? What defines the stability of the orbit? Can you calculate every possible orbit by its stability given two bodies like the Sun and Mars?

The equation for orbital velocity is vorbit = √(GM/R). So a lower orbital velocity is associated with a greater radius. So decreasing the velocity would also require moving Mars farther away from the Sun. That would take a lot of work.

My question is, take Mars, say, and install a huge rocket engine on the ground pointing out the planet (obviously) and in the opposite direction of its velocity, so that if it is turned on, the planet is slowed.

I see a big problem with this that will stop it from working
Have you figured out what it is ?

Could you explain what you mean with 'also' here? From what I know changing the velocity ís for satellites actually the way to change its orbit. What extra would you then need to do?

Sorry, I was sloppy in my thinking. The correct balance between velocity and orbit radius means that an increased radius orbit has a decreased velocity. I guess that the way to get into a higher orbit would be to momentarily increase the velocity and let the satellite break out of orbit moving to a higher radius orbit. When it comes to rest in a new orbit, it's velocity would be lower than it started at.

Sorry, I was sloppy in my thinking. The correct balance between velocity and orbit radius means that an increased radius orbit has a decreased velocity. I guess that the way to get into a higher orbit would be to momentarily increase the velocity and let the satellite break out of orbit moving to a higher radius orbit. When it comes to rest in a new orbit, it's velocity would be lower than it started at.

That's strange actually. You increase the velocity but then you end up going slower? My knowledge of orbital motions is limited, but what I seem to remember from videos about space exploration is that one way to return back to earth is to give a thrust in the opposite direction of your velocity, as to slow down. Also, debris in near orbit eventually burns in the atmosphere because it slows down due to a tiny amount of friction and falls back to earth. However, your formula indeed seems to imply the opposite... I'm confused.

Changing to a higher orbit is somewhat like throwing a ball in the air -- you have to give it some more velocity to put it up into the higher orbit, but by the time it gets there and settled into orbit it has slowed down to a lower velocity.
On the other hand, if you are in an orbit and slow down, you would begin to fall out of orbit toward the center. But in falling, your velocity increases till it gets higher than it started and can orbit at the lower height.
An elliptical orbit has some of both occurring. At the highest part of the orbit, it is going slowest. When it falls to the lowest part it picks up speed and is going fastest there. Kepler's "Law of Areas" says that a line that connects a planet to the sun sweeps out equal areas in equal times.

If you thrust in the direction opposite of the velocity vector over a short time, you will probably end up making the orbit more elliptical. Especially if you thrust when the planet is near its aphelion. It will take a huge amount of thrust. If you thrust enough, the orbit will squash into a pancake shape and Mars will fall into the Sun. But this takes an extreme amount of thrust over a short time. Perhaps a collision with another planet traveling in the opposite direction will do.

My question is, take Mars, say, and install a huge rocket engine on the ground pointing out the planet (obviously) and in the opposite direction of its velocity, so that if it is turned on, the planet is slowed.
The rocket starts to work and Mars is slowed. If the loss of velocity goes little by little, could we make Mars follow another stable orbit or would it just fall to the Sun?

If you can go more in depth: if the answer to the question asked before is yes, what equation tells me exactly how little the loss of velocity must be in order to keep the planet through another stable orbit? What defines the stability of the orbit? Can you calculate every possible orbit by its stability given two bodies like the Sun and Mars?

Thank you very much in advance!

Any orbit is "stable" as long as it doesn't intersect with the body the orbit is around, or the velocity does not equal or exceed escape velocity (technically, even then you have a stable orbit, it just won't be a closed one.) Oddly enough, it would actually take a larger velocity change to cause Mars to collide with the Sun than it would to boost it to escape velocity so that it left the solar system entirely.

Another issue with the rocket mounted on the surface, besides the rotation of the planet, that no matter how huge it is, unless you use the right type of rocket, you won't get any net change in the planet's orbit. The trick is that you need a rocket engine with an exhaust velocity greater than the escape velocity of the planet. Otherwise, the exhaust gasses initially fly off in one direction and Mars slightly in the other. but as the gasses climb away from Mars, its gravity pulling on it slows both. Eventually they come to a stop, and fall back towards each other, ending up right back where they started.

Our best chemical rockets have exhaust velocities of about 4.5 km/sec. Mars escape velocity is about 5 km/sec. So you would need something with a better higher exhaust velocity than a chemical rocket. The problem is that the high exhaust velocity engines, like Ion rockets, produce very low thrust, and so while theoretically capable of moving Mars in its orbit, it would take forever and a day to cause any appreciable difference.

Sorry, I was sloppy in my thinking. The correct balance between velocity and orbit radius means that an increased radius orbit has a decreased velocity. I guess that the way to get into a higher orbit would be to momentarily increase the velocity and let the satellite break out of orbit moving to a higher radius orbit. When it comes to rest in a new orbit, it's velocity would be lower than it started at.

You actually need a two step process if you want your final orbit to be circular. The first boost puts you in an elliptical orbit with a perihelion at the original orbt distance and an aphelion at the new orbit distance. When it reaches the new orbit distance, it will indeed be moving slower than when it started, but it will also be moving too slow to maintain a circular orbit at its new distance. Left by itself, it will start to fall back in towards its old orbit picking up speed again and will just keep orbiting between these two extremes. In order to achieve a circular orbit at this new distance, you will need to provide a another velocity boost to increase your velocity to that needed for the circular orbit. Your final orbital speed will end up being less than it was in your old orbit. Thus you will have made two velocity increases and end up with a lower final velocity.

Any orbit is "stable" as long as it doesn't intersect with the body the orbit is around, or the velocity does not equal or exceed escape velocity (technically, even then you have a stable orbit, it just won't be a closed one.) Oddly enough, it would actually take a larger velocity change to cause Mars to collide with the Sun than it would to boost it to escape velocity so that it left the solar system entirely.

That seems to make the half-escape velocity significant -- sort of a "better to continue than to go back" velocity. Have there been any practical applications of this fact?

Another issue with the rocket mounted on the surface, besides the rotation of the planet, that no matter how huge it is, unless you use the right type of rocket, you won't get any net change in the planet's orbit. The trick is that you need a rocket engine with an exhaust velocity greater than the escape velocity of the planet. Otherwise, the exhaust gasses initially fly off in one direction and Mars slightly in the other. but as the gasses climb away from Mars, its gravity pulling on it slows both. Eventually they come to a stop, and fall back towards each other, ending up right back where they started.

Good point. Interesting.

Our best chemical rockets have exhaust velocities of about 4.5 km/sec. Mars escape velocity is about 5 km/sec. So you would need something with a better higher exhaust velocity than a chemical rocket. The problem is that the high exhaust velocity engines, like Ion rockets, produce very low thrust, and so while theoretically capable of moving Mars in its orbit, it would take forever and a day to cause any appreciable difference.

And yet, we have sent an unmanned spacecraft beyond the Solar System. Does that mean that our current rockets could not do that without tricks like "slingshotting" around another planet?
EDIT: No. I see that spacecraft are entirely different. Whereas gas would fall back to Mars, the escape velocity from a spaceship is so low as to be negligible.

That seems to make the half-escape velocity significant -- sort of a "better to continue than to go back" velocity. Have there been any practical applications of this fact?Good point. Interesting.
And yet, we have sent an unmanned spacecraft beyond the Solar System. Does that mean that our current rockets could not do that without tricks like "slingshotting" around another planet?
EDIT: No. I see that spacecraft are entirely different. Whereas gas would fall back to Mars, the escape velocity from a spaceship is so low as to be negligible.

Right. Now a chemical rocket on the Moon would theoretically work, as the Moon's escape velocity is only ~2.4 km/sec. However, fighting the Moon's gravity will reduce the effective exhaust velocity of the rocket to some extent. Taking this into account, you would need about 11.5% the Moon's own mass in fuel to give the Moon enough of a boost to reach escape velocity. To give you an idea of how much this is, all the oceans of the Earth mass a little under 2% that of the Moon's . So even if you could magically transport all the water on the Earth to the Moon and then separated it into hydrogen and oxygen to burn in your rocket, you wouldn't have enough to boost the Moon entirely out of its orbit around the Earth.